- Email: [email protected]
The restoration of radiograph with an improved power spectrum equalizer
NDT&E International 38 (2005) 497–500 www.elsevier.com/locate/ndteint
The restoration of radiograph with an improved power spectrum equalizer* Wang Li-ming*, Zhao Ying-liang, Han Yan Department of Engineering and Information Science, North University of China, Taiyuan 030051, China Received 26 November 2004; revised 8 January 2005; accepted 9 January 2005 Available online 8 February 2005
Abstract Scattering is the main factor which influences the sensitivity of X-ray radiographic system. After studying the mechanism of image degradation, analyzing the advantages and disadvantages of power spectrum equalizer, and combining with the power spectrum estimation of the typical images, the authors put forward an improved restoration method on power spectrum equalizer. The practical results show that this method can depress the blur of the image, and effectively preserve the edge informations of radiograph. q 2005 Elsevier Ltd. All rights reserved. Keywords: Image restoration; Power spectrum equalizer; Radiograph; Point spread function
1. Introduction In the industrial application of X-ray for nondestructive test, the quality of radiograph is certainly important to the inspection sensitivity. One of the primary factors to impact the quality of radiograph is scattering. The scattered X-ray produces a fog and reduces the radiograph contrast and the density resolution in its turn. It also induces the blur of the image edge and reduces the spatial resolution. Especially in the radiograph of complicated workpiece with high energy X-ray, sometimes, the influence of scattered X-ray is much greater than that of the main X-ray and induces the depression and degradation of the image quality, and badly influences the detecting sensitivity. Therefore, it is important to restore the main X-ray information from the degraded image for increasing the sensitivity of the X-ray radiograph system. In the practical X-ray radiographic test, the transfer characteristic of the imaging system will also degrade the quality of the image. On the aspect of the whole system, the degradation of the image quality induced by the process of scattering and the imaging system must be both
considered to be restored. The key of image restoration is to construct an effective degradation model by which can be restored the original image.
2. The degradation of radiograph Generally, the degradation process of an image can be expressed as [1–3] gðx; yÞ Z Hf ðx; yÞ
(1)
where f(x,y) represents the original image of the workpiece projection and g(x,y) is the degraded image, operator H is described as the transform function of all degradation factors. Usually the degradation process of radiograph is thought as the output of ideal image by a linear timeinvariant system. If the digital image is denoted as f ðx; yÞ Z
ðð
dðx K x 0 ; y K y 0 Þf ðx 0 ; y 0 Þdx 0 dy 0
(2)
*
This work was supported in part by The National Natural Science Foundation of China (NSFC) under Grant 60372073. * Corresponding author. Tel.: C86 351 3557444; fax: C86 351 3557396. E-mail address: [email protected] (W. Li-ming). 0963-8695/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ndteint.2005.01.001
and the radiographic system is assumed as a linear timeinvariant one, then
498
W. Li-ming et al. / NDT&E International 38 (2005) 497–500
Fig. 1. The model of image degradation.
ðð gðx; yÞ Z
hðx K x 0 ; y K y 0 Þf ðx 0 ; y 0 Þdx 0 dy 0
Z f ðx; yÞ hðx; yÞ
Fig. 3. The ideal and practical gray-level distributions of the image.
(3)
convolution function, we have: ðð gðx; yÞ Z f ðx K x 0 ; y K y 0 Þhðx 0 ; y 0 Þdx 0 dy 0 C nðx; yÞ
h(x,y) is called as point spread function (PSF). In the practical radiograph system the noise is unavoidable and is usually thought as additive noise n(x,y), so the degradation can be shown as Fig. 1, that is
According to the elementary definition of power spectrum the autocorrelation function of g(x,y) is calculated and its Fourier transform is carried out, we obtained:
gðx; yÞ Z f ðx; yÞ hðx; yÞ C nðx; yÞ
Sgg ðu; vÞ Z Sfg ðu; vÞHðu; vÞ C Snn ðu; vÞ
(4)
In order to intuitively know the degradation of the X-ray radiograph, the X-ray radiograph of a practical stepped workpiece is shown in Fig. 2, the (a) is original image without processing and (b) is the image by contrast stretching. The workpiece is composed of many steel stairs of various thickness. Fig. 3 is the gray-level distribution along descending direction with depressed noise and ideal gray-level distribution, it is seen in figure that the edges of the stair are already too blurred to be distinguished.
3. The restoration method on power spectrum equalizer
(5)
(6)
where Sfg ðu; vÞ Z Sff ðu; vÞH ðu; vÞ
(7)
Suppose the impulse response of restoration filter is m(x,y) and the corresponding transfer function is Mðu; vÞ:f^ðx; yÞ represents the image after restoration. The images before and after restoration should meet with the following formula: ^ vÞ Z Gðu; vÞMðu; vÞ Fðu;
(8)
So the power spectrum of the image restored can be expressed as Sf^f^ ðu; vÞ Z Sff ðu; vÞjHðu; vÞj2 jMðu; vÞj2 C Snn ðu; vÞjMðu; vÞj2 (9)
3.1. The principle of the restoration method on power spectrum equalizer The presupposition of a power spectrum equalizer is that the image and noise all belongs to homogeneous random field, the noise is irrelevant to the image and the mean of noises is zero. According to the degradation model described in formula (4) and the commutative law of
According to the definition of power spectrum equalizer, the power spectrum of the image restored should be equal to that of original image. The transfer function of restoration filter can be deduced from formula (9) as follows: " #1=2 Sff ðu; vÞ Mðu; vÞ Z (10) Sff ðu; vÞjHðu; vÞj2 C Snn ðu; vÞ Evidently, if there exists no noise, i.e. Snn(u,v)Z0, the power spectrum equalizer is simplified as an inverse filter.
Fig. 2. The practical X-ray radiograph of stepped workpiece. (a) The original image, (b) after contrast stretched.
Fig. 4. The comparison between the Wiener filter and the power spectrum equalizer. (a) The Wiener filter, (b) the power spectrum equalizer.
W. Li-ming et al. / NDT&E International 38 (2005) 497–500
499
Using the formula (10) to restore the image can avoid the illcondition characteristic arising in inverse filter in case that H(u,v) is zero or very small. Compared with Wiener filtering, two methods of restoration are both insensitive to noise. But in the Wiener filter the response has to be forced to zero at H(u,v)/0. However, for power spectrum equalizer higher gain still exists at the extreme point, thus more fine structure of the image can be remained which is more suitable to flaw detecting. Fig. 4 is the comparison between transfer functions of the Wiener filter and the power spectrum equalizer.
the experiment is used to replace the power spectrum of original image Sff(u,v) when the original image is difficult to estimate or the prior knowledge is not available, but it is difficult to obtain a preferable effect [4,5]. In this text, the power spectrum of typical image described in Ref. [4] is used to improve the restoration method of the power spectrum equalizer and better effect is obtained. The power spectrum of typical image can be approximated as
3.2. The improved power spectrum equalizer
Sff ðuÞ Z BeKau
The power spectrum equalizer described in formula (10) can be used for the image restoration. Three parameters must first be obtained. They are H(u,v), Snn(u,v) and Sff(u,v), in which H(u,v) is the frequency response of point spread function, Snn(u,v) is the power spectrum of noise and Sff(u,v) is the power spectrum of original image. However, in the practical restoration the three parameters are difficult to determine. In some systems the point spread function of the system can be obtained by the corresponding prior knowledge of the image degradation, but in more another cases the degradation of the image is very complicated so that the point spread function obtained merely with the information on the image itself is a rough estimate of the ideal function. In addition the process is certainly complex. In conclusion, it is necessary to deduct a better restoration filter formula adaptable to the practical calculation from the formula (10). Substitute the formula (7) in Section 3.1 into the formula (6), we get the power spectrum of the degraded image
where B and a are constant, a is related to the frequency spectrum of original image. When the image is degraded and blurred, the frequency band width is narrower than that of the original image, so the selection of a should produce a power spectrum of the original image containing higher frequency component than that of the degraded image. The formula (13) is substituted to the formula (12) and the result is as follows:
Sgg ðu; vÞ Z Sff ðu; vÞjHðu; vÞj2 C Snn ðu; vÞ
(11)
Obviously, it is just the expression described by the denominator in formula (10). Introduced the formula (11) into the formula (10), we have Sff ðu; vÞ 1=2 Mðu; vÞ Z (12) Sgg ðu; vÞ In the above formula, the power spectrum equalizer is actually the square root of the power spectrum ratio of original image and the degraded image. Accordingly the power spectrum of the degraded image Sgg(u,v) can be calculated by means of the degraded image and the original image can be restored. The whole process of image restoration can be greatly simplified without using explicit point spread function. Usually the power spectrum of original image Sff(u,v) is estimated by the definition using a kind of images which is of the same statistic characteristic with the original image. But the images used for estimation is unpractical in engineering application on account of the diversity of the structure of tested objects. What is more? In order to simplify the calculation, a constant determined by
Sff ðuÞ MðuÞ Z Sgg ðuÞ
(13)
1=2
BeKau Z Sgg ðuÞ
1=2 (14)
When the above formula is used to carry through the calculation of the image restoration, not only it is unnecessary to use those images that are of the same statistical characteristic for the power spectrum estimation of original image by definition, but also it is unnecessary to replace the power spectrum of original image with the constant supposed by experiment. So the calculation of the image restoration is greatly simplified and the restoration quality can be effectively improved.
Fig. 5. The restorated result of practical image.
Fig. 6. Gray-level distribution of the degraded image and that of the corresponding restored image.
500
W. Li-ming et al. / NDT&E International 38 (2005) 497–500
Fig. 7. The restoration result of the image of a styles penetrometer. (a) The original image, (b) the restored image with a improved method.
3.3. The result of restoration by the improved method
4. Conclusion
The above-mentioned restoration method by the improved power spectrum equalizer is used for the Xray radiograph image restoration of stepped workpiece and the preferable result is obtained. Fig. 5 is the restorated result. Fig. 6 is the comparison of a raw gray-level distribution of the practical degraded image and that of the corresponding restored image. In the two figures it is intuitively seen that the restoration method by the improved power spectrum equalizer is of certain advantage in preserving the image edge, namely the high frequency details of the image. Fig. 7 is the restoration to the practical penetrometer image obtained by the testing system using the improved method. In the Fig. 7(a) is the original image of a stylus penetrometer and (b) is the image restored with the improved method. By the restoration operation it can be directly seen that the number of the seen stylus is by two more than that of the image without restoration and the sensitivity of detection is greatly enhanced.
From the deduction and description of the algorithm and the results in Figs. 5–7, it can be seen that the improved restoration method by power spectrum equalizer proposed in this text can effectively depress the blurred extent, enhance the edge information of X-ray radiograph and increase the sensitivity of X-ray radiograph. In addition, there is no special request to the hardware of X-ray radiographic system.
References [1] Richard AC. Image restoration using nonlinear optimization techniques with a knowledge based constraint. SPIE-2029 1993;P209–P26. [2] Rossman K. Point spread function, line spread function and modulation transfer function. Radiology 1969;93:257–72. [3] Alan LL, Robert AK. Scatter estimation for a digital radiographic system using convolution filtering. J Med Phys 1987;14(2):178–85. [4] Peters TM. Spatial filtering to improve transverse tomography. IEEE Trans Bio Med Eng 1974;21(3):214–9. [5] Shuyue C. Study on the improvement of X-ray digital radiograph quality. PhD Dissertation, Nanjing University of Science and Technology, December 2000.
The restoration of radiograph with an improved power spectrum equalizer* Wang Li-ming*, Zhao Ying-liang, Han Yan Department of Engineering and Information Science, North University of China, Taiyuan 030051, China Received 26 November 2004; revised 8 January 2005; accepted 9 January 2005 Available online 8 February 2005
Abstract Scattering is the main factor which influences the sensitivity of X-ray radiographic system. After studying the mechanism of image degradation, analyzing the advantages and disadvantages of power spectrum equalizer, and combining with the power spectrum estimation of the typical images, the authors put forward an improved restoration method on power spectrum equalizer. The practical results show that this method can depress the blur of the image, and effectively preserve the edge informations of radiograph. q 2005 Elsevier Ltd. All rights reserved. Keywords: Image restoration; Power spectrum equalizer; Radiograph; Point spread function
1. Introduction In the industrial application of X-ray for nondestructive test, the quality of radiograph is certainly important to the inspection sensitivity. One of the primary factors to impact the quality of radiograph is scattering. The scattered X-ray produces a fog and reduces the radiograph contrast and the density resolution in its turn. It also induces the blur of the image edge and reduces the spatial resolution. Especially in the radiograph of complicated workpiece with high energy X-ray, sometimes, the influence of scattered X-ray is much greater than that of the main X-ray and induces the depression and degradation of the image quality, and badly influences the detecting sensitivity. Therefore, it is important to restore the main X-ray information from the degraded image for increasing the sensitivity of the X-ray radiograph system. In the practical X-ray radiographic test, the transfer characteristic of the imaging system will also degrade the quality of the image. On the aspect of the whole system, the degradation of the image quality induced by the process of scattering and the imaging system must be both
considered to be restored. The key of image restoration is to construct an effective degradation model by which can be restored the original image.
2. The degradation of radiograph Generally, the degradation process of an image can be expressed as [1–3] gðx; yÞ Z Hf ðx; yÞ
(1)
where f(x,y) represents the original image of the workpiece projection and g(x,y) is the degraded image, operator H is described as the transform function of all degradation factors. Usually the degradation process of radiograph is thought as the output of ideal image by a linear timeinvariant system. If the digital image is denoted as f ðx; yÞ Z
ðð
dðx K x 0 ; y K y 0 Þf ðx 0 ; y 0 Þdx 0 dy 0
(2)
*
This work was supported in part by The National Natural Science Foundation of China (NSFC) under Grant 60372073. * Corresponding author. Tel.: C86 351 3557444; fax: C86 351 3557396. E-mail address: [email protected] (W. Li-ming). 0963-8695/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ndteint.2005.01.001
and the radiographic system is assumed as a linear timeinvariant one, then
498
W. Li-ming et al. / NDT&E International 38 (2005) 497–500
Fig. 1. The model of image degradation.
ðð gðx; yÞ Z
hðx K x 0 ; y K y 0 Þf ðx 0 ; y 0 Þdx 0 dy 0
Z f ðx; yÞ hðx; yÞ
Fig. 3. The ideal and practical gray-level distributions of the image.
(3)
convolution function, we have: ðð gðx; yÞ Z f ðx K x 0 ; y K y 0 Þhðx 0 ; y 0 Þdx 0 dy 0 C nðx; yÞ
h(x,y) is called as point spread function (PSF). In the practical radiograph system the noise is unavoidable and is usually thought as additive noise n(x,y), so the degradation can be shown as Fig. 1, that is
According to the elementary definition of power spectrum the autocorrelation function of g(x,y) is calculated and its Fourier transform is carried out, we obtained:
gðx; yÞ Z f ðx; yÞ hðx; yÞ C nðx; yÞ
Sgg ðu; vÞ Z Sfg ðu; vÞHðu; vÞ C Snn ðu; vÞ
(4)
In order to intuitively know the degradation of the X-ray radiograph, the X-ray radiograph of a practical stepped workpiece is shown in Fig. 2, the (a) is original image without processing and (b) is the image by contrast stretching. The workpiece is composed of many steel stairs of various thickness. Fig. 3 is the gray-level distribution along descending direction with depressed noise and ideal gray-level distribution, it is seen in figure that the edges of the stair are already too blurred to be distinguished.
3. The restoration method on power spectrum equalizer
(5)
(6)
where Sfg ðu; vÞ Z Sff ðu; vÞH ðu; vÞ
(7)
Suppose the impulse response of restoration filter is m(x,y) and the corresponding transfer function is Mðu; vÞ:f^ðx; yÞ represents the image after restoration. The images before and after restoration should meet with the following formula: ^ vÞ Z Gðu; vÞMðu; vÞ Fðu;
(8)
So the power spectrum of the image restored can be expressed as Sf^f^ ðu; vÞ Z Sff ðu; vÞjHðu; vÞj2 jMðu; vÞj2 C Snn ðu; vÞjMðu; vÞj2 (9)
3.1. The principle of the restoration method on power spectrum equalizer The presupposition of a power spectrum equalizer is that the image and noise all belongs to homogeneous random field, the noise is irrelevant to the image and the mean of noises is zero. According to the degradation model described in formula (4) and the commutative law of
According to the definition of power spectrum equalizer, the power spectrum of the image restored should be equal to that of original image. The transfer function of restoration filter can be deduced from formula (9) as follows: " #1=2 Sff ðu; vÞ Mðu; vÞ Z (10) Sff ðu; vÞjHðu; vÞj2 C Snn ðu; vÞ Evidently, if there exists no noise, i.e. Snn(u,v)Z0, the power spectrum equalizer is simplified as an inverse filter.
Fig. 2. The practical X-ray radiograph of stepped workpiece. (a) The original image, (b) after contrast stretched.
Fig. 4. The comparison between the Wiener filter and the power spectrum equalizer. (a) The Wiener filter, (b) the power spectrum equalizer.
W. Li-ming et al. / NDT&E International 38 (2005) 497–500
499
Using the formula (10) to restore the image can avoid the illcondition characteristic arising in inverse filter in case that H(u,v) is zero or very small. Compared with Wiener filtering, two methods of restoration are both insensitive to noise. But in the Wiener filter the response has to be forced to zero at H(u,v)/0. However, for power spectrum equalizer higher gain still exists at the extreme point, thus more fine structure of the image can be remained which is more suitable to flaw detecting. Fig. 4 is the comparison between transfer functions of the Wiener filter and the power spectrum equalizer.
the experiment is used to replace the power spectrum of original image Sff(u,v) when the original image is difficult to estimate or the prior knowledge is not available, but it is difficult to obtain a preferable effect [4,5]. In this text, the power spectrum of typical image described in Ref. [4] is used to improve the restoration method of the power spectrum equalizer and better effect is obtained. The power spectrum of typical image can be approximated as
3.2. The improved power spectrum equalizer
Sff ðuÞ Z BeKau
The power spectrum equalizer described in formula (10) can be used for the image restoration. Three parameters must first be obtained. They are H(u,v), Snn(u,v) and Sff(u,v), in which H(u,v) is the frequency response of point spread function, Snn(u,v) is the power spectrum of noise and Sff(u,v) is the power spectrum of original image. However, in the practical restoration the three parameters are difficult to determine. In some systems the point spread function of the system can be obtained by the corresponding prior knowledge of the image degradation, but in more another cases the degradation of the image is very complicated so that the point spread function obtained merely with the information on the image itself is a rough estimate of the ideal function. In addition the process is certainly complex. In conclusion, it is necessary to deduct a better restoration filter formula adaptable to the practical calculation from the formula (10). Substitute the formula (7) in Section 3.1 into the formula (6), we get the power spectrum of the degraded image
where B and a are constant, a is related to the frequency spectrum of original image. When the image is degraded and blurred, the frequency band width is narrower than that of the original image, so the selection of a should produce a power spectrum of the original image containing higher frequency component than that of the degraded image. The formula (13) is substituted to the formula (12) and the result is as follows:
Sgg ðu; vÞ Z Sff ðu; vÞjHðu; vÞj2 C Snn ðu; vÞ
(11)
Obviously, it is just the expression described by the denominator in formula (10). Introduced the formula (11) into the formula (10), we have Sff ðu; vÞ 1=2 Mðu; vÞ Z (12) Sgg ðu; vÞ In the above formula, the power spectrum equalizer is actually the square root of the power spectrum ratio of original image and the degraded image. Accordingly the power spectrum of the degraded image Sgg(u,v) can be calculated by means of the degraded image and the original image can be restored. The whole process of image restoration can be greatly simplified without using explicit point spread function. Usually the power spectrum of original image Sff(u,v) is estimated by the definition using a kind of images which is of the same statistic characteristic with the original image. But the images used for estimation is unpractical in engineering application on account of the diversity of the structure of tested objects. What is more? In order to simplify the calculation, a constant determined by
Sff ðuÞ MðuÞ Z Sgg ðuÞ
(13)
1=2
BeKau Z Sgg ðuÞ
1=2 (14)
When the above formula is used to carry through the calculation of the image restoration, not only it is unnecessary to use those images that are of the same statistical characteristic for the power spectrum estimation of original image by definition, but also it is unnecessary to replace the power spectrum of original image with the constant supposed by experiment. So the calculation of the image restoration is greatly simplified and the restoration quality can be effectively improved.
Fig. 5. The restorated result of practical image.
Fig. 6. Gray-level distribution of the degraded image and that of the corresponding restored image.
500
W. Li-ming et al. / NDT&E International 38 (2005) 497–500
Fig. 7. The restoration result of the image of a styles penetrometer. (a) The original image, (b) the restored image with a improved method.
3.3. The result of restoration by the improved method
4. Conclusion
The above-mentioned restoration method by the improved power spectrum equalizer is used for the Xray radiograph image restoration of stepped workpiece and the preferable result is obtained. Fig. 5 is the restorated result. Fig. 6 is the comparison of a raw gray-level distribution of the practical degraded image and that of the corresponding restored image. In the two figures it is intuitively seen that the restoration method by the improved power spectrum equalizer is of certain advantage in preserving the image edge, namely the high frequency details of the image. Fig. 7 is the restoration to the practical penetrometer image obtained by the testing system using the improved method. In the Fig. 7(a) is the original image of a stylus penetrometer and (b) is the image restored with the improved method. By the restoration operation it can be directly seen that the number of the seen stylus is by two more than that of the image without restoration and the sensitivity of detection is greatly enhanced.
From the deduction and description of the algorithm and the results in Figs. 5–7, it can be seen that the improved restoration method by power spectrum equalizer proposed in this text can effectively depress the blurred extent, enhance the edge information of X-ray radiograph and increase the sensitivity of X-ray radiograph. In addition, there is no special request to the hardware of X-ray radiographic system.
References [1] Richard AC. Image restoration using nonlinear optimization techniques with a knowledge based constraint. SPIE-2029 1993;P209–P26. [2] Rossman K. Point spread function, line spread function and modulation transfer function. Radiology 1969;93:257–72. [3] Alan LL, Robert AK. Scatter estimation for a digital radiographic system using convolution filtering. J Med Phys 1987;14(2):178–85. [4] Peters TM. Spatial filtering to improve transverse tomography. IEEE Trans Bio Med Eng 1974;21(3):214–9. [5] Shuyue C. Study on the improvement of X-ray digital radiograph quality. PhD Dissertation, Nanjing University of Science and Technology, December 2000.